Scanning Microscopy Volume 1990 Number 4 Fundamental Electron and Ion Beam Interactions with Solids for Microscopy, Article 5 Microanalysis, and Microlithography 1990 Plasmons in Scanning Transmission Electron Microscopy Electron Spectra R. H. Ritchie University of Tennessee A. Howie Cavendish Laboratory P. M. Echenique Universidad Pais Vasco G. J. Basbas Physical Review Letters T. L. Ferrell University of Tennessee SeeFollow next this page and for additional additional works authors at: https:/ /digitalcommons.usu.edu/microscopy Part of the Biology Commons Recommended Citation Ritchie, R. H.; Howie, A.; Echenique, P. M.; Basbas, G. J.; Ferrell, T. L.; and Ashley, J. C. (1990) "Plasmons in Scanning Transmission Electron Microscopy Electron Spectra," Scanning Microscopy: Vol. 1990 : No. 4 , Article 5. Available at: https://digitalcommons.usu.edu/microscopy/vol1990/iss4/5 This Article is brought to you for free and open access by the Western Dairy Center at DigitalCommons@USU. It has been accepted for inclusion in Scanning Microscopy by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. Plasmons in Scanning Transmission Electron Microscopy Electron Spectra Authors R. H. Ritchie, A. Howie, P. M. Echenique, G. J. Basbas, T. L. Ferrell, and J. C. Ashley This article is available in Scanning Microscopy: https://digitalcommons.usu.edu/microscopy/vol1990/iss4/5 Scanning Microscopy Supplement 4, 1990 (Pages 45-56) 0892-953X/90$3.00+ .00 Scanning Microscopy International, Chicago (AMF O'Hare), IL 60666 USA PLASMONS IN SCANNING TRANSMISSION ELECTRON MICROSCOPY ELECTRON SPECTRA R.H. llitchie,• A. Howie, 1 P. M. Echenique, 2 G. J. Basbas,3 T. L. Ferrell, and J. C. Ashley Oak llidge National Laboratory, P. 0. Box 2008, Oak Ridge, TN 37831-{3123 and Department of Physics, University of Tennessee, Knoxville, TN 37996 USA !Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK 2Universidad Pais Vasco, Facultad de Quimica, Aptdo 1072, San Sebastian, SPAIN 3Physical Review Letters, Box 1000, Ridge, New York 11961 USA Abstract Introduction A general self-energy formulation of the Surface and bulk plasmon excitations in condensed interaction between an electron in a scanning matter by swift charged particles have been much studied transmission electron microscope (STEM) and a localized for the last four decades [1-{3]. Here we describe the target is given. We prove a theorem relating the excitation of such collective modes by swift electrons that probability of energy transfer to that calculated may be formed into a beam incident on a localized target. classically. Local dielectric theory of target excitation for Using the finely focused probe in the scanning various geometries is discussed. The problem of transmission microscope (STEM), data can be collected localization of initially unlocalized excitations in the with a spatial resolution of 2 nm or better in the region of valence band of solids is treated by transforming cross surfaces, interfaces, small particles, and other features of sections differential in momentum transfer into inhomogeneous specimens. Energy analysis of the dependence on an impact parameter variable. We are inelastically scattered electrons gives much useful thereby able to account for experimental data in scanning information about the target. Such interactions involve electron microscopy (SEM) that show high spatial the wave-particle duality of the electrons in an resolution. interesting context. We introduce a generalized self-energy formulation of the electron-target interaction that describes the full quanta! properties of the probe. A general theorem relating the energy losses by an electron microprobe to those experienced by classical electrons with the same energy is described [7]. Recent progress in analyzing STEM data on energy losses in inhomogeneous targets using classical theory is reviewed. We also treat some aspects of secondary electron emission (SE) from such targets, emphasizing the localization of initially unlocalized excitations and the spatial distribution of collective modes created in the valence band of a solid. The Self-Ener~y in STEM It is convenient to use the self-energy concept in describing the interaction of a fast electron with an inhomogeneous target [8-13]. The essential quanta! Key Words: Plasmons, scanning transmission electron properties of the incident electron are properly treated, microscopy, inelastic interactions, electron energy losses, while the target is represented in terms of its response quanta! self-energy, localization, spatial resolution, dielectric function. Information about the spatial dependence of theory, scanning electron microscopy the interaction probability may be inferred readily from the self-energy function. A general treatment of inelastic and elastic interactions in STEM has been given [14,15]. Here we use a different emphasis couched a priori in a mixed space-energy representation. This treatment is particularly appropriate to those interactions arising in * Address for Correspondence: STEM, where information about energy transfers to localized regions of space is of interest. Here we neglect R. H. Ritchie relativistic effects and consider excitations of a target Oak Ridge National Laboratory with characteristic energies >>kT, where T is the Post Office Box 2008 temperature of the target. Oak Ridge, TN 37831-{3123 Consider the Green function G E(r,r') describing Telephone Number: (615) 574-{3208 the propagation of an electron with energy E from the 45 R.H. Ritchie, et al. Figure l (a) Figure 2 (b) (c) Figure 1. The Feynman diagram corresponding to the Figure 2. Feynman diagrams representing higher-order first-order self-energy of an electron interacting with a contributions to the self-energy of an electron. Figure 2a target as expressed in Eq. 2 corresponds to Eq. 5. space point r to the point r'. Many-body perturbation ,i,•(r') ,i,(r') theory [16,17] may be invoked to write a Dyson integral v(r) = e2 d3r' ----- , (4) equation for GE(r,r') in terms of Gi(r,r'), the J I r - r' I noninteracting Green function; for Coulomb interactions between electrons in the target. GE(r,r') = G~(r,r') We neglect elastic scattering for the present purposes. 3 3 The Feynman diagram of Fig. 1 illustrates the + f d rJ d r 2 GE(r,r 2) ~\(r 2,r1) Gi(r 1,r'). (1) interaction corresponding to Eq. 2. A term of next higher order in the electron-target interaction may be written, The spatially-dependent, proper self-energy, EE(r,r' ), of the propagating electron due to interaction with the target, in turn may be expressed as an infinite series, the first term of which may be written (5) The exact linear response function, or polarization propagator, for disturbances in the many-particle target corresponding to the Feynman diagram of Fig 2a. is given by Figures 2b and 2c illustrate two other contributions to EEof this same order. Note that the self-energy diagram "{ (Olv(r)I n) (n I v(r')IO) of Fig. 2c is obtained if we replace G by GO in Eqs. 2 and W E(r,r') = kJ 5. It does not correspond to the "proper" self-energy, n ~n + E + ia found algebraically in a straightforward manner for translationally invariant systems, and discussed in the literature (see Refs. 10,11). Generalization to still higher-order terms is straightforward [11,12]. (Olv(r')ln)(nlv(r)ID)} Analytical solution of the Dyson equation (Eq. 1 +--------- ) (3) above) is not possible for systems without translational ~n - E + ia invariance. Here we approximate the self-energy of a projectile interacting with an arbitrary target by a series of terms that are ordered according to the number of where a is a positive infinitesimal, ( I 0), In)) is the exact times the projectile interacts with the target. In doing so state vector of the many-particle target in the (ground, we use the linear response function of the target since we nth excited) state and 6'n = 6' - in, where ( ~' 6'n) is expect that this procedure will yield reasonable results in 0 0 many cases. the energy of the (ground, nth excited) state. The The electron Green function may be approximated interaction energy v(r) may be expressed in terms of in the standard form (,i,•(r), ,i,(r)), the wave (creation, annihilation) operator tor a particle at position r in the target, viz., (6) 46 Plasmons in STEM Electron Spectra where uk(r) is an exact eigenfunction for the fully SOURCE interacting electron, Ek is the eigenenergy of this state and ois a positive infinitesimal. If the electron is prepared in a state specified by the state function 1f;(r), then the "averaged" self-energy 0 ELECTRON LENS is complex . Its real part yields the shift in the electron's energy while the imaginary part, when multiplied by 2/'h, gives the rate at which it is scattered out of the initial state. E0 is understood here to be the energy of the state corresponding to 1f;0. Equation 7 may be simplified by usin~ Eq. 6 for GE(r,r') and by approximating the exact EElr,r') by its lowest-order form Ei(r,r') from Eq.2. Then Figure 3. A schematic diagram representing a typical STEM configuration. The target is located at impact parameter b with respect to the focused beam and the 1f;*(r) uk(r)uk(r'* )1f; (r') energy analyzer subtends the half-angle 0m at the angle 0 0 x ------------WE 1 (r,r') . (8) 0D with respect to the incident beam. E - Ek - E' + i o 0 straightforward way from Eq. 7 in lowest order when one This is a generalization of an expression given in [11] for takes E 1_ Using the Lehmann form, Eq.3, for W, 0 1f;*(r)uk(r)uk(r')'I/J * (r') 3 3 0 0 El= EEJ d rf d r' -------- E exp[ik•(r-r')]/13 o k n E k + E n + i 6 (11) 0 0 k [E-Ek + io] x ( 0 I v( r) I n) ( n I v( r' ) IO) .